My Math Forum Symmetry properties of graphs

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 July 31st, 2015, 01:17 AM #1 Newbie   Joined: Jun 2015 From: South Africa Posts: 27 Thanks: 0 Symmetry properties of graphs In each of the following cases find the least positive value of α for which: a) cos(α-θ)˚=sinθ˚ b) sin(α-θ)˚=cos(α+θ)˚ There are a whole bunch more... I did a) easily because obviously cos(90-θ)˚=sinθ˚, so α = 90. I'm struggling to work out b) though... Thanks for any help.
 July 31st, 2015, 04:50 PM #2 Math Team   Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 This is how I would do it. Simplifying $\sin(\alpha - \theta) = \cos(\alpha + \theta)$, we get $$\sin(\alpha - \theta) = \cos(\alpha + \theta)\\ \sin\alpha\cos\theta - \sin\theta\cos\alpha = \cos\alpha\cos\theta - \sin\alpha\sin\theta\\ \sin\alpha(\cos\theta + \sin\theta) = \cos\alpha(\cos\theta + \sin\theta)\qquad\qquad\qquad(1)$$ In cases where $\cos\theta + \sin\theta = 0$, equality holds trivially. Otherwise, $(1)$ can be simplified to $$\sin\alpha = \cos\alpha,$$ or, $$\tan\alpha = 1.$$ We now see that the answer is $\alpha = 45^\circ$

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